562 research outputs found
What Moser Could Have Asked: Counting Hamilton Cycles in Tournaments
Moser asked for a construction of explicit tournaments on vertices having
at least Hamilton cycles. We show that he could have asked
for rather more
Factors of sums and alternating sums involving binomial coefficients and powers of integers
We study divisibility properties of certain sums and alternating sums
involving binomial coefficients and powers of integers. For example, we prove
that for all positive integers , , and any
nonnegative integer , there holds {align*} \sum_{k=0}^{n_1}\epsilon^k
(2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod
(n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any
nonnegative integer and positive integer such that is odd, where .Comment: 14 pages, to appear in Int. J. Number Theor
Challenges of socio-economically evaluating wildfire management in and adjacent to non-industrial private forestland in the Western United States
Non-industrial private forests (NIPFs) in the United States generate many unpriced benefits for both the landholder and society generally. These values can be enhanced or diminished by wildfire management in situ and on adjacent public land. This paper considers the problem of accommodating non-marketed NIPF values affected by wildfire in social benefit-cost analysis to evaluate the efficiency of fire suppression activities. There are substantial gaps in scientific understanding about how the spatial and temporal provision of non-market values are affected by wildfire, and considerable challenges in evaluating social welfare change arising from specific wildfire events. This presents serious impediments to adapting price-based decision-support tools to incorporate non-market values. Departure from the historic range and variability of ecological conditions is proposed as an alternative framework to support wildfire management decisions in and adjacent to NIPF when non-market values are important
Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers
For all nonnegative integers n, the Franel numbers are defined as We confirm two conjectures of Z.-W. Sun on
congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0
\pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2
\pmod{p^5}, where n is a positive integer and p>3 is a prime.Comment: 8 pages, minor changes, to appear in Integral Transforms Spec. Func
Almost Odd Random Sum-Free Sets
We show that if S_1 is a strongly complete sum-free set of positive
integers, and if S_0 is a finite sum-free set, then with positive probability a random
sum-free set U contains S_0 and is contained in S_0\cup S_1. As a corollary we show that
with positive probability, 2 is the only even element of a random sum-free set
Elliptic curves, modular forms, and sums of Hurwitz class numbers
Let H(N) denote the Hurwitz class number. It is known that if is a prime,
then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper,
we investigate the behavior of this sum with the additional condition . Three different methods will be explored for determining the values
of such sums. First, we will count isomorphism classes of elliptic curves over
finite fields. Second, we will express the sums as coefficients of modular
forms. Third, we will manipulate the Eichler-Selberg trace for ula for Hecke
operators to obtain Hurwitz class number relations. The cases and 4 are
treated in full. Partial results, as well as several conjectures, are given for
and 7.Comment: Preprint of an old pape
Linear Momentum Density in Quasistatic Electromagnetic Systems
We discuss a couple of simple quasistatic electromagnetic systems in which
the density of electromagnetic linear momentum can be easily computed. The
examples are also used to illustrate how the total electromagnetic linear
momentum, which may also be calculated by using the vector potential, can be
understood as a consequence of the violation of the action-reaction principle,
because a non-null external force is required to maintain constant the
mechanical linear momentum. We show how one can avoid the divergence in the
interaction linear electromagnetic momentum of a system composed by an
idealization often used in textbooks (an infinite straight current) and a point
charge.Comment: 22 pages, 5 figures, to appear in Eur. J. Phy
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